return the coefficient of a specific power in an Ore polynomial - Maple Help

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OreTools[Utility][Coefficient] - return the coefficient of a specific power in an Ore polynomial

OreTools[Utility][Coefficients] - return the coefficient sequence of an Ore polynomial

OreTools[Utility][Degree] - return the degree of an Ore polynomial with respect to the noncommutative indeterminate

OreTools[Utility][LeadingCoefficient] - return the leading coefficient of an Ore polynomial

OreTools[Utility][LowDegree] - return the degree of the least power with nonzero coefficient

OreTools[Utility][RandOrePoly] - return a random Ore polynomial

OreTools[Utility][TrailingCoefficient] - return the trailing coefficient of an Ore Polynomial

OreTools[Utility][VariableDegree] - return the maximal degree of the coefficients of an Ore Polynomial in the variable in an Ore algebra

Calling Sequence

Coefficient(Poly, n)

Coefficients(Poly)

Degree(Poly)

LeadingCoefficient(Poly)

LowDegree(Poly)

RandOrePoly(A, opts)

TrailingCoefficient(Poly)

VariableDegree(Poly, A)

Parameters

Poly

-

Ore polynomial; to define an Ore polynomial, see OreTools/OrePoly

n

-

non-negative integer

A

-

Ore algebra

opts

-

options

Description

• 

The Coefficient(Poly, n) calling sequence returns the coefficient of the nth power of the noncommutative indeterminate in Poly.

• 

The Coefficients(Poly) calling sequence returns the sequence of coefficients of Poly.

• 

The Degree(Poly) calling sequence returns the degree of Poly with respect to the noncommutative indeterminate.

• 

The LeadingCoefficient(Poly)] calling sequence returns the leading coefficient of Poly.

• 

The LowDegree(Poly) calling sequence returns the trailing degree of Poly.

• 

The RandOrePoly(A) calling sequence returns a random Ore polynomial in the Ore algebra A.

  

The first argument A specifies the ring in which the polynomial is to be generated. The possible options are:

  

coeffs - Generate the coefficients

  

terms - Number of terms in the noncommutative indeterminate

  

degree - Degree on the noncommutative indeterminate

• 

The TrailingCoefficient(Poly) calling sequence returns the trailing coefficient of A.

• 

The VariableDegree(Poly, A) calling sequence returns the maximal degree of coefficients of Poly with respect to the variable in A.  Note that the coefficients of Poly are supposed to be polynomials in the variable.

• 

For a brief review of pseudo-linear algebra (also known as Ore algebra), see OreAlgebra.

Examples

withOreTools:

withOreTools[Utility]:

Poly:='OrePoly'0,2n1,0,1n

Poly:=OrePoly0,2n1,0,1n

(1)

CoefficientPoly,1

2n1

(2)

CoefficientPoly,6

0

(3)

CoefficientsPoly

0,2n1,0,1n

(4)

DegreePoly

3

(5)

Degree'OrePoly'0

∞

(6)

LeadingCoefficientPoly

1n

(7)

LowDegreePoly

1

(8)

LowDegree'OrePoly'0

∞

(9)

TrailingCoefficientPoly

2n1

(10)

TrailingCoefficient'OrePoly'0

0

(11)

A:=SetOreRingn,'shift':

Poly:=RandOrePolyA

Poly:=OrePoly72n5+37n423n3+87n2+44n+29,50n5+23n4+75n392n2+6n+74,17n575n410n37n240n+42,10n5+62n482n3+80n244n+71,62n4+97n373n24n83,7n5+22n455n394n2+87n56

(12)

DegreePoly

5

(13)

VariableDegreePoly,A

5

(14)

VariableDegree'OrePoly'0,A

∞

(15)

B:=SetOreRingx,'differential':

C:=RandOrePolyB,coeffs=polynomdegree=3,terms=2,terms=2,degree=10

C:=OrePoly0,0,0,0,40x381x,11+95x

(16)

VariableDegreeC,B

3

(17)

F:=SetOreRingx,q,'qshift':

UtilityRandOrePolyF,coeffs=ratpolydegnum=1,degden=2,terms=2,degree=5,terms=3,degree=10

OrePoly0,87q+47x908848q,0,16q+30x2772qx96,0,0,0,0,51q+77x+9528qx+55q

(18)

See Also

OreTools, OreTools/OreAlgebra, OreTools/SetOreRing, OreTools[Utility], randpoly


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